The hierarchical nature of galaxy formation in the ΛCDM framework often leads to the formation of multiple supermassive black holes (SMBHs) in the galactic nuclei. The timescale over which galaxies merge, plays a crucial role in shaping the dynamical evolution and the merger dynamics of their central SMBHs, which may result in the formation of a triple (or multiple) SMBH system. While the evolution of binary SMBH is well studied, the long-term dynamics of triple SMBH systems, particularly in non-spherical potentials, still remain less understood. In this work, we investigate the impact of triaxial shape of galaxies in the dynamical evolution of triple SMBHs with initial conditions drawn from the ROMULUS25 cosmological simulation, using different high-resolution gravitodynamical N-body simulations. We explore different orbital configurations and host galaxy shapes, tracking the orbital evolution from a galactic inspiral phase to the formation of hard binaries at sub-parsec separation, and use the observed hardening rates to project the time of coalescence. In all cases, the two most massive SMBHs form a rapidly hardening binary that coalesces within a fraction of a Hubble time, while the third black hole either forms a stable hierarchical triple system with the heavier binary, or remains on a wide orbit.